INEQUALITIES

Transcription

1 INEQUALITIES Once the meaning of a solution is understood, it can be applied to understanding solutions of inequalities and sstems of inequalities. Inequalities tpicall have infinitel man solutions, and students learn was to represent such solutions. For additional information see the Math Notes boes in Lessons and Eample 1 Solve each equation or inequalit below. Eplain what the solution for each one represents. Then eplain how the equation and inequalities are related to each other. Solution: = < There are man was to solve the equation, including graphing, factoring and using the Zero Product Propert, or using the Quadratic Formula. Factoring and use the Zero Product Propert to solve is shown below = 0 ( 5)( + 1) = 0 = 5, = 1 Check: If = 5: (5) 2 4(5) 5 = = 0 ü If = 1: ( 1) 2 4( 1) 5 = = 0 ü The second quadratic is an inequalit. To solve this inequalit, utilize a number line to emphasize what the solution represents. Start b solving the related quadratic equation (which was done above) = 0 The soluions to the equatoin are = 5 and = 1. B placing these two points on a number line, the act as boundar points, dividing the number line into three regions. Since the original inequalit is strictl less than, use open circles for the boundar points Choose an number in each of the regions to see if the number will make the original inequalit true or false. Solutions will make the inequalit true. Note: You onl need to check one point in each region. Solution continues on net page CPM Educational Program. All rights reserved. Core Connections Integrated III

2 Chapter 3 Solution continued from previous page. Choosing a point in each region and substituting in into the inequalit gives: = 2 = 0 = 7 ( 2) 2 4( 2) 5? < 0 (0) 2 4(0) 5 <? 0 (7) 2 4(7) 5? < <? <? ? < 0 7 <? 0 5 <? 0 16 <? 0 false true false Highlight the region on the number line that makes the inequlit true, as shown at right. This solution can also be represented algebraicall as 1 < < The last inequalit in the eample has a. Having both and means an -coordinate graph needs to be used to show the solutions. Graph the parabola using a solid line because the original inequalit is greater then or equal to. The graph of the parabola at right divides the plane into two regions: the part within the bowl of the parabola the interior and the region outside the parabola. The points on the parabola represent where = Test a point from one of the regions to check whether it will make the inequalit true or false. As before, we are looking for the true region. The point (0, 0) is an eas point to use. 0? (0) 2 4(0) 5 0? ? 5 True! Therefore the region containing the point (0, 0) is the solution. This means an point chosen in this region, the bowl of the parabola, will make the inequalit true. To illustrate that this region is the solution, we shade this region of the graph. To see how these equations and inequalities are related, eamine the graph of the parabola. Where are the zeros? Where are the -values of the parabola negative? The zeros are the -intercepts on the graph, or = 1 and = 5, which ou determined b solving the equation. The graph is negative when it dips below the -ais, and this happens when is between 1 and 5. Solving the first inequalit answered this question as well. Therefore, the graph could have answered the first two parts quickl. Parent Guide and Etra Practice 2015 CPM Educational Program. All rights reserved. 35

3 Eample 2 Han and Lea have been building jet roamers and pod racers. Each jet roamer requires one jet pack and three crstallic fuel tanks, while each pod racer requires two jet packs and four crstallic fuel tanks. Han and Lea s suppliers can onl produce 100 jet packs and 270 fuel tanks each week, and due to manufacturing conditions, Han and Lea can build no more than 30 pod racers each week. Each jet roamer makes a profit of 1 tig (their form of currenc) while each pod racer makes a profit of 4 tigs. a. If Han and Lea receive an order for 12 jet roamers and 22 pod racers, how man of each part will the need to fill this order? If the can fill this order, how man tigs will the make? b. Write a list of constraints, an inequalit for each constraint, and sketch a graph showing all inequalities with the points of intersection labeled. How man jet roamers and pod racers should Han and Lea build to maimize their profits? Solution: This problem is an eample of a linear programming problem, and although the name might conjure up images of computer programming, these problems are not done on a computer. This problem can be solved b creating a sstem of inequalities that, when graphed, creates a feasibilit region. This region contains the solution for the number of jet roamers and pod racers Han and Lea should make to maimize their profit. a. Jet packs: 1(12) + 2(22) = = 56 Fuel tanks: 3(12) + 4(22) = = 124 Each result is within the constraints, so it is possible for Han and Lea to fill this order. If the do, the will make 1(12) + 4(22) = = 100 tigs. b. Begin b defining the variables. Let = the number of jet roamers Han and Lea will make, and = the number of pod racers. 0, and 0 because a negative number of items cannot be produced. A jet roamer requires one jet pack while a pod racer requires two. There are onl 100 jet packs available each week, so Each jet roamer requires three crstallic fuel tanks and each pod racer requires four. This translates into the inequalit since onl 270 fuel tanks are available each week. Lastl, since Han and Lea cannot make more than 30 pod racers, we can write 30. These inequalities are all shown on the graph at right. The region common to all constraints is shaded. This is the feasibilit region because choosing a point in this shaded area gives ou a combination of jet roamers and pod racers that Han and Lea can produce under the given restraints. Pod Racers Solution continues on net page Jet Roamers CPM Educational Program. All rights reserved. Core Connections Integrated III

4 Chapter 3 Solution continued from previous page. The equation used to calculate the profit is P = To maimize profits, test all the vertices of the feasibilit region in the profit equation. These points are (0, 0), (0, 30), (40, 30), (70, 15), and (90, 0). (0, 0): P = 1(0) + 4(0) = 0 (0, 30): P = 1(0) + 4(30) = 120 (40, 30): P = 1(40) + 4(30) = 160 (70, 15): P = 1(70) + 4(15) = 130 (90, 0): P = 1(90) + 4(0) = 90 The greatest profit is 160 tigs when Han and Lea build 40 jet roamers and 30 pod racers. Problems Graph the following sstem of inequalities. 1. < > < < 3 4 ( 1)2 + 6 > < 10 + > 4 < 2 > < 12 > ( + 1) < ( + 2) 3 > For each of the following problems, write a sstem of inequalities that when graphed will produce the shaded region Parent Guide and Etra Practice 2015 CPM Educational Program. All rights reserved. 37

5 9. Ramon and Thea are considering opening their own business. The wish to make and sell alien dolls the call Hauteans and Zotions. Each Hautean sells for $1.00 while each Zotion sells for $1.50. The can make up to 20 Hauteans and 40 Zotions, but no more than 50 dolls total. When Ramon and Thea go to cit hall to get a business license, the find there are a few more restrictions on their production. The number of Zotions (the more epensive item) can be no more than three times the number of Hauteans (the cheaper item). How man of each doll should Ramon and Thea make to maimize their profit? What will the profit be? 10. Sam and Emma are plant managers for the Stick Chew Cand Compan that specializes in delectable gourmet candies. Their two most popular candies are Chocolate Chews and Peanut and Jell Jimmies. Each batch of Chocolate Chews takes 1 teaspoon of vanilla while each batch of the Peanut and Jell Jimmies uses two teaspoons of vanilla. The have at most 20 teaspoons of vanilla on hand as the use onl the freshest of ingredients. The Chocolate Chews use two teaspoons of baking soda while the Peanut and Jell Jimmies use three teaspoons of baking soda. The onl have 36 teaspoons of baking soda on hand. Because of production restrictions, the can make no more than 15 batches of Chocolate Chews and no more than 7 batches of Peanut and Jell Jimmies. Sam and Emma have been given the task of determining how man batches of each cand the should produce if the make $3.00 profit for each batch of Chocolate Chews and $2.00 for each batch of Peanut and Jell Jimmies. Help them out b writing the inequalities described here, graphing the feasibilit region, and determining their maimum profit. Answers CPM Educational Program. All rights reserved. Core Connections Integrated III

6 Chapter ( 6) The graph of the feasibilit region is shown at right. The inequalities are 0, 0, + 50, 20, 40, and 50 3, where = number of Hauteans and = number of 40 Zotions. The profit is given b P = Maimum A profit seems to occur at point A (12.5, 37.5), but there is a problem with this point. Ramon and Thea cannot make a half of a doll (or at least that does not seem possible). Tr these nearb points: (12, 37), (12, 38), (13, 37), and 10 (13, 38). The point that gives maimum profit and is still in the feasibilit region is (13, 37). The should make 13 Hautean and 37 Zotion dolls for a profit of $ The graph of the feasibilit region is shown at right. The inequalities are 0, 0, 7, 15, , and , where = number of Chocolate Chews and = number of Peanut and Jell Jimmies. The profit is given b P = The point that seems to give the maimum profit is (15, 2.5) but this onl works if half batches can be made. Instead, choose the point (15, 2) which means Sam and Emma should make 15 batches of Chocolate Chews and 2 batches of Peanut and Jell Jimmies. Their profit will be $ Parent Guide and Etra Practice 2015 CPM Educational Program. All rights reserved. 39